Optimal query complexity for fractional quantum evolution

Abstract: Given oracle access to an unknown unitary $U=e<sup>{iH}$ , the fractional query problem asks how many queries are required to implement a noninteger power $U<sup>t=e<sup>{itH}$, $0<t<1$, when the spectrum is separated from the branch cut by a gap $δ$. Quantum singular value transformation gives an upper bound of $O!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$ queries for approximation error $\varepsilon$. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any $N$-query circuit to the approximation of $e<sup>{itθ}$ by a trigonometric polynomial with degree bounded by $O(N)$, together with Remez inequality. This allows us to establish the lower bound of $$Ω<em>τ!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$$. Consequently, the optimal query complexity for fractional query problem is $$Θ</em>τ!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$$, showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a $Ω_τ!\left(\log\frac{1}{\varepsilon}\right)$ bound uniform to $δ$.